Wide-field optical imaging of electrical charge and chemical reactions at the solid–liquid interface

Significance The solid–liquid interface is an important entity across disciplines. The ability to characterize interfacial properties can prove decisive in a range of vital application areas including pharmaceutical formulations, catalysts and energy production, and storage devices. The electrical charge that develops at the solid–liquid interface exerts a profound influence on surface dynamics and interactions, and yet we have few, if any, broadly applicable techniques that permit interfaces and their properties to be interrogated in a straightforward fashion. Here we demonstrate the ability to perform simple, wide-field optical imaging of surface electrical charge at the spatial resolution of the optical microscope. This fosters large-area (millimeter scale), direct visualization of surface chemistry and time-dependent changes due to chemical reactions at surfaces.


Interferometry-based height measurement in the lens-coverglass system
The height in the gap created between lens and coverglass was measured using the interferometric pattern (Newton's rings) created by reflection of excitation laser light from the two flanking surfaces.
This pattern was recorded on a CCD camera (Thorlabs) as shown in Fig. 1A. First, the centre of the interference pattern was found by binarizing the image using increasing grey level thresholds, and subsequently finding the centre of mass of the radially symmetric binarized pattern for each increasing threshold value, as previously described by Morrin et al. (Fig. S1B)(1). The true centre of mass of the image is estimated as the mean value of the estimated centre of mass for all increasing thresholds. The centre coordinates are then used to take a radial average over the image to generate a radial intensity trace, ( ), which is typically a trace from the centre ( = 0) to the edge of the field of view ( = 130 μm) containing 3 intensity minima and 3 maxima, (red and green symbols overlaid on trace in Fig. S1B).
The variable-height gap in our measurements consists of a well-defined multi-layer geometry, glass/water/glass ( Fig. 1-2) and glass/thin film/water/glass ( Fig. 3-4). We used the Fresnel equations to calculate the reflection and transmission coefficients at each interface. Since our system is illuminated with collimated light, we assumed normal incidence for these calculations.
The optical transfer matrix method was used to calculate the reflected intensity with respect to the height of the water-filled gap. (2) Fig. S1B displays minima and maxima for a calculated interference pattern as a function of height of the system. The calculated intensity from each minimum to the following maximum can be mapped to the measured intensity trace in a piecewise fashion in order to determine the height as a function of radial location in the system. If is defined as the transfer matrix between two materials with refractive index and and is the propagation matrix within the material whose index is , then the total transfer matrix can be written as = where is the refractive index for medium 1 (glass), and is the refractive index for medium 2 (water). To extract the intensity of the reflected light, we use the solution for the reflection coefficient = , where the indices identify the element in the 2×2 propagation matrix.
The transmission coefficient can be obtained from the following equation The final solution for is given by: Finally, the reflection coefficient is determined in the same way as described above: We generally only consider two interfaces (a glass/water and water/silica) interface in our experiments. We include an additional interface when the experiments include metal oxide films

Poisson-Boltzmann numerical model for surface potential and surface charge density determination
The values of surface electrical potential and charge density reported in this work were obtained by matching the experimentally measured radial intensity distributions of the light-emitting probes ( ( )) with theoretical intensity profiles calculated using the Poisson-Boltzmann (PB) equation for the variable height gap. In this section, we describe the analysis procedure using the PB equation to determine the theoretically expected probe molecule concentration profiles in the system.
Importantly we also use optical corrections to account for fluorescent dye emission coupling into the substrate surfaces in order to arrive at a theoretical ( ) profile under a given set of experimental conditions.
The PB equation can be expressed in dimensionless form as: where = is the dimensionless electrical potential and = is the Debye length.
Here is the vacuum permittivity, the permittivity of the medium, is the Boltzmann constant, is the temperature, is the Avogadro number, is the salt concentration in solution, and is the elementary charge. We specify the boundary conditions of the problem, which can be either a Dirichlet boundary condition of fixed potential at the surfaces, or a Neumann boundary condition which entails a constant electric field normal to the surface. We apply a Neumann boundary condition at the surfaces, which can be expressed as where is the surface normal pointing into the gap. Eq. (S6) corresponds to a constant surface charge density, , at the bounding surfaces.
Radial symmetry and the short length scales of our experimental setup facilitate rapid computation of the electrical potential distribution ( ( , )) in the gap, described by the NLPB equation. The gap created by the lens in the experiment is cylindrically symmetric around the contact region with the bottom substrate, allowing us to reduce the dimension of the simulation to a 2D axisymmetric model ( Fig. S3A). We can further reduce the dimensionality of the system under the assumption that the curvature of the lens is very small. This assumption is valid because within the radial distance covered in the field of view (0 μm < < 110 μm), we measured a gap height between 0 and 500 nm. Hence, the solution for the electrical potential distribution in the gap can be approximated by solving the NLPB in 1D for each height thus enabling us to reconstruct the solution for the entire axisymmetric geometry based on a series of 1-D solutions at variable (Fig. S3B). We further illustrate this reduction in dimensionality in Fig. S3, where both the 2D axisymmetric and 1D potential distributions were solved in COMSOL and found to be identical within numerical precision ( Fig. S3C-D).
Having obtained the potential distribution ( , ) within the gap between the lens and coverglass, we proceeded to calculate the concentration of the charged probe molecules in the gap. The local concentration of probe (fluorescent dye) molecules in the gap can be described by the Boltzmann distribution. Assuming a bulk probe concentration , the spatial varying probe concentration can be written as: where is the effective charge of the probe molecule and ( , ) is the electrical potential distribution in the gap (5)(6)(7)(8)(9)(10). The experimentally measured image of the dye emission is radially symmetric yielding a radially dependent intensity ( ) measurement. The intensity measured in the image plane corresponds to a 2-D projection of the 3-D dye distribution in the gap. Integrating Eq.
(S7), we obtain an expression for the theoretically expected radial intensity profile given as follows: We use a proportionality to relate the expected intensity ( ) and the dye concentration in order to account for other contributions to the intensity from the experimental setup that we do not directly measure. These contributions include those from the quantum efficiency of the dye and camera sensor, collection efficiency of the objective, the point spread function of the optical system, optical losses within the set-up, etc. For this work, we assume these contributions remain constant over the field of view and that they contribute linear scaling factors outside the integral. Note that because Eq. (S8) is solely a function of and the charge of the probe, , the solution of the PB equation directly permits us to calculate the expected intensity (within a factor given by the system dependent proportionality constant ). For typical measurement conditions (from 0.01 mM to 1 mM NaCl) remained fairly constant at ≈ 0.2 ±0.01 for both pH 6 and pH 9. Varying the value of the surface charge density in Eq. (S6), we generate theoretical ( ) curves for a probe molecule of a known charge, and then compare the calculated profiles with the experimental measurements. We are thus able to determine the value of the unknown surface charge density and the corresponding electrical potential characterizing the surface material of interest.

2.A. Accounting for super-critical angle fluorescence contributions in the collected optical intensity
In order to accurately model the expected intensity profile due to the distribution of fluorescent probe molecules in the gap, we consider the coupling of photons from a dipole close to the two water-glass interfaces on either side of the solution-filled region. In this sub-section, we describe an optical collection function, OCF( ′), that must be incorporated into Eq. (S8). The primary purpose of the OCF is that it accurately accounts for the total emission collected from the dye which is strongly influenced by the presence of the dielectric discontinuity (water/glass interface) at the upper surface of the gap (Fig. 1A, Fig. S4A). Eq. (S8) is thus modified to yield where ′ = ( ) − is the distance of the emitter from the upper glass/water interface. We calculate the OCF of our system as follows. An excited dipole in a medium of refractive index , which is at a distance < from an interface with a medium of larger refractive index > (where is the emission wavelength), will emit Supercritical Angle Fluorescence (SAF), i.e., emission into surface at angles above the critical angle of total internal reflection ≈ 61° at the water-glass interface. This emission occurs in addition to fluorescence at angles below the critical angle (Undercritical Angle Fluorescence, UAF). Here is the angle included between a ray emanating from the dipole emitter and the normal vector at the surface substrate ( Fig. S4A-D) (11,12). For dipoles located close to the interface, approximately two-thirds of the total fluorescence is emitted into the higher index medium, as the evanescent field of the dipole emitters couples into the denser medium and is transformed into propagating plane waves. These rays propagate at or beyond the critical angle (13). Importantly, the amplitude of the SAF plane waves depends exponentially on the height of the dipole, ′, above the layer. Thus, for dipoles more than about a wavelength from the surface there will be virtually no light coupled into directions beyond the critical angle.

2.A.i. Radiating dipole near an interface
Enderlein et al. (14) describe the angular emission into the water half-space for a dipole for both vertical and parallel orientations with respect to the plane of the interface as follows: Similarly, for emission into the glass half-space we have The total emission into either half space can be calculated by integrating over a solid angle: where is the total emitted power into one of the half spaces, with = 1 and 2 denoting water and glass respectively.
For a radiating dipole with a random orientation we then have: In the experiment we collect emission up to collection angles given by the numerical aperture (NA) of the objective. We therefore integrate over an interval of the polar angle, = sin , yielding an Optical Collection Function (OCF)

2.A.ii. Estimation of the collected emission in the experimental setup.
We indicated in the previous section that the OCF primarily accounted for influences on the dipole emission arising from the "upper" surface of the gap. We discuss here why distance of the emitter from the lower half space does not strongly influence the total collected photons. In Fig. S4A,C-D we see that for dipoles close to an interface, the emission into the glass does increase dramatically with distance. However, this increase in emission arises from light propagating due to the evanescent field of the emitter, which is directed entirely at angles equal to and greater than critical angle. The emission into the undercritical angle region on the other hand remains unaltered.
Therefore, in experiments such as ours where NA<1.33, the collection of emission from the lower dielectric half space has no distance dependence, as indicated by evaluating Eq. (S12) and (S13) in Fig. S4A-D. In other words, in the regime ≤ ≈ 61 ∘ , we only collect UAF emission which is independent of the location of the emitter given by .
On the other hand, the upper dielectric discontinuity does exert a strong influence on the total collection. This is because the emission into the lower water slab is in fact affected by the presence of the interface and also carries a strong distance dependence, which is true for all collection angles ( Fig. S4A).
In our present model, we treat the contribution from the two interfaces (upper glass surface/water and water/coverglass) independently and superpose them in the final result. A more comprehensive model of collected photons in a system containing multiple dielectric interfaces can be developed with the help of finite difference simulations and will be discussed in future work. In future work, the potential distribution calculations with the PB equation could be extended to go beyond the constant potential or constant charge model and include charge regulation, finite ion size, and other contributions to the total interaction e.g., hydration forces. (15)(16)(17) 3. Extracting the chemical species properties of the silica surface using measurements of surface potential at pH 4.5, 6 and 9

2.B. Comparing calculated ( ) profiles with experimentally measured intensity distributions.
Silica surfaces acquire electrical charge primarily through the dissociation of silanol (SiOH) groups in contact with an electrolyte, as illustrated by the following reaction: Silica surfaces are thought to be characterized by at least three different types of ionizable species i with characteristic acid dissociation constants p whose extent of deprotonation, , depends on the solution pH (18)(19)(20)(21). In order to understand the charging process responsible for the surface potentials measured in our experiments (Fig. 2), we generalize the method reported by Behrens and Grier to calculate the electric charge density of silica surfaces in contact with aqueous electrolytes (22). In general, we can write the total charge density at the silica surface as follows: where e is the elementary charge, is the number of silanol groups per surface area and is the fractional preponderance of the species, with ∑ = 1. The ionisation probability in turn can be written as: = 1 1 + 10 ( ) (S19) where = ± 1 denotes the sign of the charge of the surface group in the ionised state (15). We note that depends on pH and the dimensionless surface (or local) electrical potential , which can significantly influence the dissociation process. This phenomenon is known as charge regulation (15). Eq. (S18) and (S19) describe the dependence of the surface charge on the surface electrical potential when chemical reactions govern the charging process. The surface charge also depends on surface electrical potential via the PB equation, a model that describes the screening of the surface potential by counterions in the electrolyte. An exact solution of the NLPB equation for surface charge density as a function of the surface electrical potential of an isolated planar surface is given by the Grahame equation: Our aim in this section is to describe how to determine values for the surface density of ionizable groups (silanol) , as well as the p and the fractional preponderance of the ionizable species using opto-electrostatic measurements of the surface charge density, , at various pH values. We simultaneously solve Eq. (S18) and (S20) to obtain an electrical surface charge , which we can compare with the obtained from the experiments for a given measurement pH and salt concentration in solution. We vary the model parameters , p and in order to minimize the difference between the measured and . The standard deviations of the measurement are included as weights in the optimization process.
We first validate the method outlined above using simulated data as inputs (Fig. S7). In Fig. S7A, the pH-dependence of is calculated for a fictitious model surface whose properties we assume are described by values p = 3, 6 and 12, as well as a fixed density of ionizable surface groups, We then apply the same data analysis procedure to surface charge density values inferred from experimental measurements. We model our experimental silica surfaces with a trimodal p distribution given by 3 types of ionizable groups whose p values are approximately 3, 6 and 9. (18,19,21) Furthermore, the total group density and number of active dissociating sites for each species depend on experimental condition, and sample treatment history. In this work we reported estimates of the total silanol group density , the fractions , and acid dissociation constants p of each species assuming that the surfaces we examine do not change their properties depending on treatment history. We use data obtained for surface charge density under 7 different experimental conditions from pH 4.5 to pH 9 and salt concentrations of 0.01, 0.1, and 1 mM as presented in Fig. S8. We used the residual minimization procedure described above to extract values of the 6 unknown parameters of interest. We obtained silanol group density of = 0.13 ± 0.01 nm -2 , as shown in Fig. S8B and C. Furthermore, within a trimodal p description we found that the data pointed to the values p ≤ 3, p = 7.2 ± 0.8 and p = 9.65 ± 0.10, where the third ( = 3) least acidic species appeared to be the most abundant with ≈ 0.90. We further found that = 0.06 ± 0.01 and = 0.04 ± 0.01 which implies that relatively small fractions of groups with p and p appear to be responsible for a non-negligible surface charge at lower pH, i.e. at conditions under which the groups at p would be expected to be completely uncharged.

3.A.
Estimating the uncertainty in the determination of , and . The uncertainty determination on each of the inferred parameters can be illustrated with the example shown in Fig. S8B-D. In each case, one of the parameters in the 3D parameter space of , and p was fixed to the value that minimized the overall deviations between the model and the measured electrical surface charge. The resulting 2D surface plots are presented in Fig. S8B, C and D, respectively, where a minimum for the remaining two parameters was found. The reported uncertainties on , and were defined as the deviation of each parameter from the mean value required to increase the total residual by 10% from the global minimum residual value.

Characterization of effective charge for fluorescent probe molecules using surfaces with a known surface potential
Having characterized the surface potential of the silica surfaces in the lens-coverglass gap as described in the main text and in SI Appendix, Sections 1-2, we used the system to characterize the effective charge of fluorescent molecules such as organic dyes and single-stranded (ss) DNA (Fig. 2D, Fig. S9). Solving for in Eq. (S9) using the known silica surface potential , we obtained measurements of estimated values for each probe molecule as shown in Fig. 2D. In the case of the organic dyes, we note that ≈ as is expected for low object charge  Fig. S6 as individual, spatially offset data points overlaid on the boxplots for each sampled FOV in each independent measurement.
6. Constructing surface electrical potential and charge density images using the scanning probe system We used the following procedure in order to convert measured fluorescent intensity distributions obtained using the scanning probe system into estimated spatial distributions of surface charge density and potential on the substrate (Figs. 5E-F). We initially assume a value of surface charge density, , for the "square hole" (silica) regions of the substrate as well as the upper platform or probe surface. Prior knowledge of this value is not necessary as described later, but for this analysis we use an initial value = −0.0075 e/nm known from previous measurements in the lens-based system for glass surfaces (Fig. 2). First, we construct simulated 2D images of intensity for a 2D surface carrying a heterogeneous charge density. Images are calculated for specific gap heights, H= 50 nm and 500 nm, which are known independently from interferometric measurement. We then apply an additive background signal level as well noise to each pixel in order to simulate the experimental situation closely. Next, we apply a Fourier filtering step to remove the low frequency components of the image, which is the same as the procedure used to process the experimentally measured images. Varying the charge density in the "test" regions (TiO2 lines) from 0.001 /nm to 0.1 /nm in the simulated images gives us a range of expected interest contrast levels for these regions with respect to the reference SiO2 surface. We then treat these contrast levels as a library or look-up-table of optical contrast values which can be directly related to surface electrical properties. Next we examine the experimentally measured images at the pixel-by-pixel level and convert these local contrast values to surface electrical potential and corresponding charge densities suggested by our look-up-table of contrasts. In a subsequent step, we note that varying the charge density of the more strongly charged silica regions, which contributes to the background intensity, permits us to simultaneously fine-tune the initial "guesstimate" values for the charge density in these regions. This is because the ratio of measured or calculated optical contrasts at two different heights depends on both the properties of the silica and TiO2 regions, as the observed intensity values are governed by Eq. (S9) and the Fourier filtering step only permits us to measure the intensity increment relative to the SiO2 background. Note that this data analysis approach, exploiting the optical contrast between two different regions of the substrate, is specific to this type of experiment. In general, measurements of the intensity distribution at multiple different gap heights will support extraction of pixel-wise charge density values without recourse to referencing to a particular region of the substrate.

Simulations of minimum feature size determination using opto-electrostatic imaging
In order to explore the minimum feature size that could be discerned by means of our optoelectrostatic imaging technique, we performed simulations of the expected signal-to-noise ratio (SNR). Here we define SNR as the ratio between difference in fluorescence signal between two regions, e.g., the TiO2 features and the SiO2 substrate, and the noise on the SiO2 region. The may be used to enhance optical contrast due to electrostatics (Fig. S10B-C). To further explore this property, we evaluate the SNR of a simulated 500 nm TiO2 feature on glass at 0.1 mM salt concentration ( ≈ 30 nm) and vary the gap height of the system. We find a maximum SNR for all probes occurs at a height ≈ 110 nm, with a significant increase in contrast for highly charged probes. We note a 4-fold increase of the SNR going from a probe with a single charge -1 to -8 charges. If we require the minimum resolvable feature size to have an SNR of 2, then Fig.   S10C shows that a dimension of at least ≈2κ is required. Interestingly a saturation of the SNR is achieved with a feature radius > 200 nm (independent of the probe charge), assuming a fixed gap height of 110 nm. We attribute this result to the feature size exceeding both the Debye length and the width of the point spread function.
These simulations can be extended to estimate the optical contrast arising from heterogenous patterns of arbitrary thin film materials, using the surface potential values estimated from homogeneous thin film measurements for various pH conditions (Fig. 3). Here contrast is defined as the difference between the intensity of a less charged surface (appearing brighter due to greater local dye concentration) and a highly charged surrounding surface (appearing darker due to a low dye concentration). Our experimental results showed that the optical contrast due to electrostatics between TiO2 (grid) and SiO2 (substrate) is high at pH≈6 and low at pH≈9 (Fig. 4 and Fig. 5) using a dye of charge = − 3 (Atto 542c). We simulated the expected SNR for a TiO2 grid on a SiO2 substrate (Fig. S11A-B) using the surface potential values measured in homogenous thin film substrates (Fig. 3). The simulation (Fig. S11B) showed that a significant contrast (SNR ≈ 4) between TiO2 and SiO2 is observed at pH 6 and reduced contrast is observed at pH 9 (SNR ≈ 2) using a dye of -3 charge (Atto 542c), which agrees qualitatively with our experimental results (Fig. Fig 5E-H).

4A-C and
Finally, we examine the measurement of small electrical surface potentials in Fig. S11C-E. The SNR analysis outlined above was repeated for features whose surfaces were at a small potential difference of Δ = ± 0.5 with respect to the surrounding substrate surface. We observed the maximum achievable SNR in such systems does not increase even with the use of highly charged probe molecules. However, increasing probe charge causes the gap height at which the peak in the SNR occurs to shift to larger values. This means that although the ability to optically detect small differences in surface potentials is not significantly improved, highly charged probes would permit the detection of a similar level of image contrast at larger gap heights or effectively larger . We further note that the contrast (SNR) is larger for negative values of Δ .